Quasi-Cores in bargaining sets

We propose a nonempty-valued subsolution of the Mas-Colell Bargaining Set on the class of TU games satisfying grand coalition zero-monotonicity, a weaker condition than superadditivity, zero-monotonicity and balancedness. The subsolution is a slight modification of the Shapley-Shubik Quasi-Core. The Zhou Bargaining Set is a refinement of the Mas-Colell Bargaining Set. We also give a nonempty-valued subsolution of the Zhou Bargaining Set on the class of all TU games satisfying grand coalition superadditivity, a weaker condition than superadditivity and balancedness. This subsolution is a modification of the Zhou Bargaining Set which is described by excesses. These results are examined for TU games with coalition structures.